The Fundamental Theorem of Calculus
Let v(t) be a velocity function on the time interval [a,b]. The function v(t) describes the movement of something–maybe a car, maybe an emu, maybe a banana slug. The banana slug is at some starting position when t = a, travels some distance from t = a to t = b, and is at some ending position when t = b.
If v(t) ≥ 0 on [a,b], then is positive and is the distance travelled between time t = a and time t = b.
We could also say
If v(t) ≤ 0 on [a,b], then is negative and is the distance travelled between time t = a and t = b, but in the "opposite" direction.
In other words,
If v(t) is sometimes negative and sometimes positive on [a,b], then is the distance travelled in the positive direction
minus the distance travelled in the negative direction
We can also describe this by
Regardless of whether the velocity function is positive, negative, or a little bit of each,
We're going to introduce a new function s(t) so that we can write this equation more compactly. Let
s(t) = position at time t.
So s(a) = position at time t = a, s(b) = position at time t = b, etc.
On the interval [a,b], s(a) is the position of the car or cheetah at the start of the interval and s(b) is the position of the car or cheetah at the end of the interval. The change in position of the car or cheetah over the interval [a,b] is
(ending position) – (starting position) = s(b) – s(a)
However, the change in position of the car or cheetah over the interval [a,b] is also . This means
We'd like to point out that velocity is the derivative of position (change in position with respect to time). In symbols,
v(t) = s'(t).
For the following examples and exercises, assume s(t) is the position function and v(t) = s'(t) is the velocity function.
describes a relationship between the three values , s(a), and s(b). Most problems involving this equation will give you two of these values and ask you to find the third.
You could be given the value of the integral and one of the values s(a) or s(b).
We've been working with the equation
in this form because this is how the Fundamental Theorem of Calculus is usually given. However, we can also rearrange the equation
by adding s(a) to both sides to get this:
This rearranged equation says that if you take the starting position (s(a)) and add the change in position you get the ending position
Some problems are easier if we use the rearranged equation, especially those where we're given s(a) and and asked to find s(b).